Maximal Ideal and Variety

Suppose that is a non-zero ideal of . Define the variety associated with as .

Claim: If there exists such that then is a maximal ideal.

Failed attempt: For all from the definition of . Define the substitution homomorphism as . It suffices to prove that , for then, by Hilbert's Nullstellensatz, is a maximal ideal.

However, it doesn't seem that is true. Certainly . But there may be polynomials in that are not in it seems. For but there could be some if and there exists such that . It seems this could plausibly occur, but I haven't been able to construct such a counter example. So I am unable to prove that .

Finally, in working with this problem, I couldn't see a way to get to the structure of through . I thought maybe a contrapositive or proof by contradiction might work, but to no avail.

Suppose that is a non-zero ideal of . Define the variety associated with as .

Claim: If there exists such that then is a maximal ideal.

Failed attempt: For all from the definition of . Define the substitution homomorphism as . It suffices to prove that , for then, by Hilbert's Nullstellensatz, is a maximal ideal.

However, it doesn't seem that is true. Certainly . But there may be polynomials in that are not in it seems. For but there could be some if and there exists such that . It seems this could plausibly occur, but I haven't been able to construct such a counter example. So I am unable to prove that .

Finally, in working with this problem, I couldn't see a way to get to the structure of through . I thought maybe a contrapositive or proof by contradiction might work, but to no avail.

I have a question on your claim. If we choose , then V(I)={(0,0)}. In this case I is not a maximal ideal in C[x_1, x_2]. Anyhow, is a maximal ideal in C[x_1, x_2].